Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets $G$ is precisely the smallest 𝜎-algebra containing $G.$ It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Definition of a monotone class

A monotone class is a family (i.e. class) $M$ of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means $M$ has the following properties:

1. if $A_1,A_2,\dots \in M$ and $A_1\subseteq A_2\subseteq \cdots$ then $\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\in M,$ and
2. if $B_1,B_2,\dots \in M$ and $B_1\supseteq B_2\supseteq \cdots$ then $\bigcap _{i=1}^{\mathrm{\infty }}{B}_i\in M.$

Monotone class theorem for sets

Monotone class theorem for sets — Let $G$ be an algebra of sets and define $M(G)$ to be the smallest monotone class containing $G.$ Then $M(G)$ is precisely the 𝜎-algebra generated by $G$; that is $\sigma (G)=M(G).$

Monotone class theorem for functions

Monotone class theorem for functions — Let ${𝒜}$ be a π-system that contains $\mathrm{\Omega }$ and let ${\mathscr{H}}$ be a collection of functions from $\mathrm{\Omega }$ to $\mathbb{ℝ}$ with the following properties:

1. If $A\in {𝒜}$ then ${\mathbf{1}}_A\in {\mathscr{H}}$ where ${\mathbf{1}}_A$ denotes the indicator function of $A.$
2. If $f,g\in {\mathscr{H}}$ and $c\in \mathbb{ℝ}$ then $f+g$ and $cf\in {\mathscr{H}}.$
3. If $f_n\in {\mathscr{H}}$ is a sequence of non-negative functions that increase to a bounded function $f$ then $f\in {\mathscr{H}}.$

Then ${\mathscr{H}}$ contains all bounded functions that are measurable with respect to $\sigma ({𝒜}),$ which is the 𝜎-algebra generated by ${𝒜}.$

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.

Proof

The assumption $\mathrm{\Omega }\in {𝒜},$ (2), and (3) imply that ${𝒢}=\{A:{\mathbf{1}}_A\in {\mathscr{H}}\}$ is a 𝜆-system. By (1) and the π−𝜆 theorem, $\sigma ({𝒜})\subseteq {𝒢}.$ Statement (2) implies that ${\mathscr{H}}$ contains all simple functions, and then (3) implies that ${\mathscr{H}}$ contains all bounded functions measurable with respect to $\sigma ({𝒜}).$

Results and applications

As a corollary, if $G$ is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of $G.$ By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-algebra. The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.

See also

• Dynkin system – Family closed under complements and countable disjoint unions
• π-𝜆 theorem – Family closed under complements and countable disjoint unionsPages displaying short descriptions of redirect targets
• π-system – Family of sets closed under intersection
• σ-algebra – Algebraic structure of set algebra

References

• Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.